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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineJeanette has punch cards for her favorite tea house and her favorite coffee shop. She currently has 1010 punches on the tea punch card and 11 punch on the coffee punch card. Given her regular routine, she consistently earns 22 new punches per week on the tea punch card and 33 on the coffee punch card. Before too long, Jeanette will have the same number of punches on each card. How many punches will Jeanette have on each card? How long will that take?\newlineJeanette will have _\_ punches on each card in _\_ weeks.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineJeanette has punch cards for her favorite tea house and her favorite coffee shop. She currently has 1010 punches on the tea punch card and 11 punch on the coffee punch card. Given her regular routine, she consistently earns 22 new punches per week on the tea punch card and 33 on the coffee punch card. Before too long, Jeanette will have the same number of punches on each card. How many punches will Jeanette have on each card? How long will that take?\newlineJeanette will have _\_ punches on each card in _\_ weeks.
  1. Define variables: Let's define the variables: Let xx be the number of weeks it takes for Jeanette to have the same number of punches on both cards. Let TT be the total number of punches on the tea punch card, and CC be the total number of punches on the coffee punch card after xx weeks.
  2. Write equations: We can write two equations based on the information given. Jeanette starts with 1010 punches on the tea card and earns 22 punches per week. So, after xx weeks, she will have T=10+2xT = 10 + 2x punches on the tea card.
  3. Set equations equal: Similarly, Jeanette starts with 11 punch on the coffee card and earns 33 punches per week. So, after xx weeks, she will have C=1+3xC = 1 + 3x punches on the coffee card.
  4. Solve for x: We want to find out when TT equals CC, so we set the two equations equal to each other: 10+2x=1+3x10 + 2x = 1 + 3x.
  5. Substitute xx into equations: Now, we solve for xx. Subtract 2x2x from both sides to get: 10=1+x10 = 1 + x.
  6. Calculate punches after 99 weeks: Subtract 11 from both sides to isolate xx: 9=x9 = x.
  7. Check work: Now that we have xx, we can find out how many punches Jeanette will have on each card after 99 weeks. We substitute x=9x = 9 into either T=10+2xT = 10 + 2x or C=1+3xC = 1 + 3x.
  8. Check work: Now that we have xx, we can find out how many punches Jeanette will have on each card after 99 weeks. We substitute x=9x = 9 into either T=10+2xT = 10 + 2x or C=1+3xC = 1 + 3x.Substituting x=9x = 9 into T=10+2xT = 10 + 2x gives us T=10+2(9)=10+18=28T = 10 + 2(9) = 10 + 18 = 28.
  9. Check work: Now that we have xx, we can find out how many punches Jeanette will have on each card after 99 weeks. We substitute x=9x = 9 into either T=10+2xT = 10 + 2x or C=1+3xC = 1 + 3x.Substituting x=9x = 9 into T=10+2xT = 10 + 2x gives us T=10+2(9)=10+18=28T = 10 + 2(9) = 10 + 18 = 28.To check our work, we substitute x=9x = 9 into C=1+3xC = 1 + 3x as well: 9900.
  10. Check work: Now that we have xx, we can find out how many punches Jeanette will have on each card after 99 weeks. We substitute x=9x = 9 into either T=10+2xT = 10 + 2x or C=1+3xC = 1 + 3x.Substituting x=9x = 9 into T=10+2xT = 10 + 2x gives us T=10+2(9)=10+18=28T = 10 + 2(9) = 10 + 18 = 28.To check our work, we substitute x=9x = 9 into C=1+3xC = 1 + 3x as well: 9900.Since 9911 and 9922 are equal, we have found the correct number of punches on each card after 99 weeks. Jeanette will have 9944 punches on each card in 99 weeks.

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